Algebra PYQ — Page 56

If $a$ and $b$ are real numbers such that ${(2+\alpha )}^{4}=a+b\alpha$, where $\alpha =\frac{-1+i\sqrt{3}}{2}$, then $a+b$ is equal to:

Let $A=[\begin{matrix}x & 1 \\ 1 & 0\end{matrix}],x\epsilon R$ and ${A}^{4}=[{a}_{ij}].$ If ${a}_{11}=109,$ then ${a}_{22}$ is equal to_____________.

If the sum of the series $20+19\frac{3}{5}+19\frac{1}{5}+18\frac{4}{5}+..........$ up to ${n}^{\text{th }}$ term is $488$and the ${n}^{\text{th }}$ term is negative, then :

Set $A$ has $m$elements and set $B$ has $n$elements. If the total number of subsets of $A$ is $112$ more than the total number of subsets of $B$, then the value of $m\cdot n$ is___.

If ${p}$ denotes the fractional part of the number $p,$ then ${\frac{{3}^{200}}{8}}$ is equal to

If ${z}_{1},{z}_{2}$ are complex numbers such that $Re({z}_{1})=|{z}_{1}-1|$ and $Re({z}_{2})=|{z}_{2}-1|$ and $\mathrm{arg}({z}_{1}-{z}_{2})=\frac{\pi }{6}$, then $Im({z}_{1}+{z}_{2})$ is equal to :

The coefficient of ${x}^{4}$ in the expansion of ${(1+x+{x}^{2}+{x}^{3})}^{6}$ in powers of $x,$ is $\ldots ..$

If the constant term in the binomial expansion of ${(\sqrt{x}-\frac{k}{{x}^{2}})}^{10}$ is $405$, then $|k|$ equals :

If $\alpha$ and $\beta$ be two roots of the equation ${x}^{2}-64x+256=0.$ Then the value of ${(\frac{{\alpha }^{3}}{{\beta }^{5}})}^{\frac{1}{8}}+{(\frac{{\beta }^{3}}{{\alpha }^{5}})}^{\frac{1}{8}}$ is :

If$A=[\begin{matrix}\mathrm{cos}\theta & \text{ isin}\theta \\ \text{isin}\theta & \mathrm{cos}\theta \end{matrix}],(\theta =\frac{\pi }{24})$ and ${A}^{5}=[\begin{matrix}a & b \\ c & d\end{matrix}],$ where $i=\sqrt{-1}$, then which one of the following is not true?

If $A=(\begin{matrix}2 & 2 \\ 9 & 4\end{matrix})$ and $I=(\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}),$ then $10 {A}^{-1}$, is equal to.

If the system of linear equations $x+y+3z=0$ $x+3y+{k}^{2}z=0$ $3x+y+3z=0$ has a non-zero solution $(x,y,z)$ for some $k\in R,$ then $x+(\frac{y}{z})$ is equal to :

If ${3}^{2\mathrm{sin}2\alpha -1},14$ and ${3}^{4-2\mathrm{sin}2\alpha }$ are the first three terms of an A.P. for some $\alpha$ , then the sixth term of this A.P. is

Let $z$ be a complex number such that $|\frac{z-i}{z+2i}|=1$ and $|z|=\frac{5}{2}$ . Then, the value of $|z+3i|$ is

For a suitably chosen real constant $a$, let a function, $f:R-{-a}\rightarrow R$ be defined by $f(x)=\frac{a-x}{a+x}$. Further supposed that for any real number $x\neq -a,$and $f(x)\neq -a,(\mathrm{fof})(x)=x$. Then $f(-\frac{1}{2})$ is equal to :

Let $f:R\rightarrow R$ be a function which satisfies $f(x+y)=f(x)+f(y),\forall x,y\in R$ . If $f(1)=2$ and $g(n)=\sum _{k=1}^{(n-1)}f(k),$$n\in N$ then the value of $n$, for which $g(n)=20$, is

Let $f:(1, 3)\rightarrow R$, be a function defined by $f(x)=\frac{x[x]}{1+{x}^{2}},$ where $[x]$, denotes the greatest integer $\leq x.$ Then the range of $f$, is

The system of linear equations $\lambda x+2y+2z=5$ $2\lambda x+3y+5z=8$ $4x+\lambda y+6z=10$ has

If $x=\sum _{n=0}^{\infty }{(-1)}^{n}{\mathrm{tan}}^{2}⁡\theta$ and $y=\sum _{n=0}^{\infty }{\mathrm{cos}}^{2n}⁡\theta ,$ for $0<\theta <\frac{\pi }{4},$ then:

The product ${2}^{\frac{1}{4}}\cdot {4}^{\frac{1}{16}}\cdot {8}^{\frac{1}{48}}\cdot {16}^{\frac{1}{128}}\cdot ....$ to $\infty$ is equal to:

The sum $\sum _{k=1}^{20}(1+2+3+\ldots +k)$ is ___________.

Let ${a}_{1},{a}_{2},{a}_{3},\ldots$, be a $G.P.$ such that ${a}_{1}<0,{a}_{1}+{a}_{2}=4$ and ${a}_{3}+{a}_{4}=16$. If $\sum _{i=1}^{9}{a}_{i}=4\lambda$, then $\lambda$, is equal to.

Let $A$ be a $2\times 2$ real matrix with entries from ${0,1}$ and $|A|\neq 0$. Consider the following two statements; $(P)$ If $A\neq {l}_{2}$, then $|A|=-1$ $(Q)$ If $|A|=1$, then $tr(A)=2$ Where ${l}_{2}$ denotes $2\times 2$ identity matrix and $tr(A)$ denotes the sum of the diagonal entries of $A$. Then

The values of $\lambda$ and $\mu$ for which the system of linear equations $x+y+z=2$, $x+2y+3z=5$, $x+3y+\lambda z=\mu$ has infinitely many solutions, are respectively